# Transform normal to beta distribution

What is the simplest way to transform a normal$(\mu,\sigma)$ distribution to a beta$(a,b)$ distribution? I'm interested in knowing if there is an exact solution, but also if there are approximations that will achieve this? I'm not interested in a specific -- ie normal$(3,2)$ to beta$(0.5,3)$ -- transformation, but rather, general solutions.

• The beta distribution is bounded at both ends and the Gaussian distribution is not. How do you expect to handle that issue? – shadowtalker Oct 29 '16 at 22:52
• Charlie ... Do you know the parameters of the normal you have and of the beta you want? – Glen_b -Reinstate Monica Oct 29 '16 at 22:55
• @ssdecontrol to go from $(-\infty,\infty)$ to $(0,1)$ is trivial (there's an infinity of transformations that accomplish that -- the set of cdfs). There are much larger issues. – Glen_b -Reinstate Monica Oct 29 '16 at 22:58
• I'm sorry, can you be specific about what you mean by "general solutions"? It's not clear what problem you're asking us to solve – Glen_b -Reinstate Monica Oct 29 '16 at 22:59
• A general solution involves the CDF of the Normal distribution $F$, the CDF of the Beta distribution $G$, and an arbitrary measure-preserving function $t:(0,1)\to(0,1)$: the transformation $x\to F^{-1}(t(G(x))$ does the trick. There are loads of such functions $t$. For instance, you can break the interval $(0,1)$ into $n$ equal-length segments at the values $i/n$, pick any permutation $\sigma$ of $1,2,\ldots,n$, and map $x\in ((i-1)/n,i/n)$ to $x+(\sigma(i)-i)/n$ for each $i$. Because this is so extremely general, we would like to know more specifically what you mean by "transform." – whuber Oct 30 '16 at 15:34

If you're transforming from $X\sim N(\mu,\sigma^2)$ to $Y\sim \text{Beta}(\alpha,\beta)$, then $Y=F_Y^{-1}[\Phi(\frac{X-\mu}{\sigma})]$ where $F_Y$ is the desired cdf of $Y$ and $\Phi$ is the standard normal cdf will achieve the result and preserve ordering (e.g. if $X_1<X_2$ and you transform $X_i$ to $Y_i$, for $i=1,2$ then $Y_1<Y_2$).
• @gung I was thinking of one that specifically treats the inverse-cdf of cdf transformation (i.e. of the type $F^{-1}(G(X))$) in answering the question. There's one in particular I half remember but I haven't turned it up. – Glen_b -Reinstate Monica Oct 30 '16 at 0:52